How to simplify logarithms

Find out how to simplify logarithms by writing a logarithmic expression as a single logarithm with these exercises. 

Examples showing how to simplify logarithms

Exercise #1

Simplify log₃ 40 - log₃ 10

Using the quotient property, log₃ 40 - log₃ 10 = log₃  40 / 10  

Simplify log₃  40 / 10  to get log₃ 4

log₃ 40 - log₃ 10 = log₃ 4

Exercise #2

Simplify log₄ 3 + log₄ 6

Using the product property, log₄ 3 + log₄ 6 = log₄  3 x 6  

Simplify log₄  3 x 6  to get log₄ 18

log₄ 3 + log₄ 6 = log₄ 18

Exercise #3

Simplify log₁₀ 9 + log₁₀ 5  - log₁₀ 15     

Using a combination of the product rule and the quotient rule, we can simplify this logarithmic expression  as shown below.

log₁₀ 9 + log₁₀ 5  - log₁₀ 15   =   log₁₀ ( 9 x 5 )  - log₁₀ 15
                                               
                                                =   log₁₀ ( 45 )  - log₁₀ 15

                                                =   log₁₀ ( 45 / 15 ) 

                                                =   log₁₀ 3 

Exercise #4

Simplify log₅ 1 / 8 + 3 log₅ 4  

Using a combination of the product rule and the power rule, we can simplify as shown below.

log₅ 1 / 8 + 3 log₅ 4   =  log₅ 1 / 8 +  log₅ 4³

                                   =  log₅ 1 / 8 +  log₅ 64

                                   =  log₅ (1 / 8 x 64 )

                                   =  log₅ ( 64 / 8 )

                                   = log₅ 8

Here is yet another example clearly showing how to simplify a logarithmic expression using the properties of logarithms.

In the example below, we use the power property and the product property to simplify log₆ 24 + 2 log₆ 3. Logarithms can be simplified using only one property or a combination of all 3 properties.